Finite Group Databasebeta

← #22595387

Subgroups — #22595387

Order 9088.

560 conjugacy classes of subgroups; only the 121 normal subgroups are shown.

normal subgroups only

This group has 560 conjugacy classes of subgroups — too many to draw. Only its 121 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#17985926C212yesmaximal, complemented
2#17985912C212yesmaximal, complemented
3#17985911C211yesmaximal, characteristic, complemented
4#17985892C212yesmaximal, complemented
5#17985891C212yesmaximal, complemented
6#17985889C212yesmaximal, complemented
7#17985854C212yesmaximal
8#17985854C211yesmaximal, characteristic
9#397×C71C211yesmaximal, characteristic, complemented
10C24×D71C2211yescharacteristic
11C22×Dic142C2211yescharacteristic, complemented
12C2×C4⋊Dic71C2212yescomplemented
13C2×C22⋊Dic71C2211yescharacteristic
14C2×C22⋊Dic71C2212yes—
15C2×(C2×D71)⋊C4C2212yes—
16C2×(C2×D71)⋊C4C2212yes—
17C2×(C2×D71)⋊C4C2212yes—
18C2×(C2×D71)⋊C4C2212yes—
19C2×(C2×D71)⋊C4C2212yes—
20C2×(C2×D71)⋊C4C2212yes—
21C2×C4×Dic71C2212yescomplemented
22C71⋊#95C2212yescomplemented
23C71⋊#95C2212yescomplemented
24C71⋊#95C2211yescharacteristic, complemented
25C71⋊#95C2212yescomplemented
26C2×C4⋊C4×C71C2211yescharacteristic, complemented
27C2×C22⋊C4×C71C2212yes—
28C2×C22⋊C4×C71C2211yescharacteristic
29#95×C71C2212yescomplemented
30#95×C71C2211yescharacteristic, complemented
31C23×D71—22no—
32C23×D71—24no—
33C23×D71—41no—
34C23×D71—42no—
35C23×D71—41no—
36C23×D71C2312yes—
37C2×Dic142—41no—
38C2×Dic142—42no—
39C22×Dic71C2312yes—
40C22×Dic71C2312yes—
41C22×Dic71C2312yes—
42C4⋊Dic71—42no—
43C22⋊Dic71—41no—
44C22⋊Dic71—42no—
45(C2×D71)⋊C4—24no—
46(C2×D71)⋊C4—24no—
47(C2×D71)⋊C4—42no—
48(C2×D71)⋊C4—42no—
49(C2×D71)⋊C4—42no—
50(C2×D71)⋊C4—42no—
51C4×Dic71—42no—
52C24×C71C2311yescharacteristic
53C22×C4×C71C2311yescharacteristic
54C22×C4×C71C2312yes—
55C22×C4×C71C2311yescharacteristic
56C22×C4×C71C2312yes—
57C4⋊C4×C71—41no—
58C22⋊C4×C71—24no—
59C22⋊C4×C71—41no—
60C23×C71—22no—
61C23×C71—21no—
62C23×C71—21no—
63C23×C71—22no—
64C23×C71#4112yes—
65C23×C71C2411yescharacteristic
66C22×D71—22no—
67C22×D71—22no—
68C22×D71—22no—
69C22×D71—22no—
70C22×D71—22no—
71C22×D71—22no—
72C22×D71—44no—
73C22×D71—48no—
74C22×D71—48no—
75C22×D71—44no—
76C22×D71—44no—
77C22×D71#4114yes—
78C2×C4×C71—22no—
79C2×C4×C71—22no—
80C2×C4×C71—22no—
81C2×C4×C71—22no—
82C2×C4×C71—21no—
83C2×C4×C71—21no—
84C2×C4×C71—22no—
85C2×C4×C71—22no—
86C2×C4×C71—22no—
87C2×C4×C71—22no—
88C2×Dic71—22no—
89C2×Dic71—22no—
90C2×Dic71—22no—
91C2×Dic71—22no—
92C2×Dic71—22no—
93C2×Dic71—22no—
94C2×Dic71—22no—
95C2×Dic71—22no—
96C2×Dic71—22no—
97Dic142—44no—
98C22×C71—22no—
99C22×C71—24no—
100C22×C71—42no—
101C22×C71—41no—
102C22×C71—41no—
103C22×C71C2×#4112yes—
104C22×C71C2×#4111yescharacteristic
105C22×C71H(5,𝔽2)11yescharacteristic
106C22×C71H(5,𝔽2)11yescharacteristic
107C22×C71H(5,𝔽2)12yes—
108C2×D71—24no—
109C2×D71—24no—
110C2×D71—24no—
111C2×D71—416no—
112C2×D71—42no—
113C2×D71—42no—
114C2×D71—42no—
115C2×D71—42no—
116C4×C71—42no—
117C4×C71—42no—
118C4×C71—41no—
119C4×C71—41no—
120Dic71—42no—
121Dic71—42no—
122Dic71—42no—
123C2×C71—42no—
124C2×C71#53811yescharacteristic
125C2×C71#53812yes—
126C2×C71#55112yes—
127C2×C71#55111yescharacteristic
128C2×C71H(3,𝔽4)11yescharacteristic
129D71—44no—
130#2736—711nomaximal, Sylow
131C71#273611yescharacteristic, complemented, Sylow
132#397—712no—
133#397D7111yescharacteristic, complemented
134#393—711no—
135#393—712no—
136#388—712no—
137#388—712no—
138#388—712no—
139#379—711no—
140#379—712no—
141C25—711no—
142C22×Q8—711no—
143C2×C4⋊C4—712no—
144C2×C4⋊C4C2×D7111yescharacteristic, complemented
145C2×C22⋊C4—711no—
146C2×C22⋊C4—712no—
147C2×C22⋊C4—712no—
148C2×C22⋊C4—712no—
149C2×C22⋊C4—712no—
150C2×C22⋊C4—712no—
151C2×C22⋊C4—712no—
152C2×C22⋊C4—712no—
153C2×C22⋊C4C2×D7112yes—
154C2×C22⋊C4C2×D7111yescharacteristic
155C2×C42—712no—
156#95—712no—
157#95—711no—
158#95—712no—
159#95—712no—
160#95C2×D7112yescomplemented
161#95C2×D7111yescharacteristic, complemented
162C24—2841no—
163C24—2841no—
164C24—2842no—
165C24—712no—
166C24—1424no—
167C24—1422no—
168C24C22×D7111yescharacteristic
169C2×Q8—2841no—
170C2×Q8—2842no—
171C22×C4—712no—
172C22×C4—712no—
173C22×C4—712no—
174C22×C4C22×D7111yescharacteristic
175C22×C4C22×D7112yes—
176C22×C4C22×D7111yescharacteristic
177C22×C4C22×D7112yes—
178C4⋊C4—41no—
179C4⋊C4—2842no—
180C22⋊C4—24no—
181C22⋊C4—41no—
182C22⋊C4—2842no—
183C22⋊C4—2842no—
184C22⋊C4—2842no—
185C22⋊C4—2842no—
186C22⋊C4—2841no—
187C22⋊C4—2842no—
188C22⋊C4—1424no—
189C22⋊C4—1424no—
190C42—2842no—
191C23—22no—
192C23—21no—
193C23—21no—
194C23—22no—
195C23—2844no—
196C23—2848no—
197C23—2844no—
198C23—2848no—
199C23—2844no—
200C23—714no—
201C23—1422no—
202C23—1422no—
203C23—1422no—
204C23—1422no—
205C23—1422no—
206C23—1422no—
207C23#1176586412yes—
208C23C23×D7111yescharacteristic
209Q8—2844no—
210C2×C4—22no—
211C2×C4—22no—
212C2×C4—22no—
213C2×C4—22no—
214C2×C4—21no—
215C2×C4—21no—
216C2×C4—22no—
217C2×C4—22no—
218C2×C4—22no—
219C2×C4—22no—
220C2×C4—1422no—
221C2×C4—1422no—
222C2×C4—1422no—
223C2×C4—1422no—
224C2×C4—1422no—
225C2×C4—1422no—
226C2×C4—1422no—
227C2×C4—1422no—
228C2×C4—1422no—
229C22—22no—
230C22—24no—
231C22—42no—
232C22—41no—
233C22—41no—
234C22—28416no—
235C22—2842no—
236C22—2842no—
237C22—2842no—
238C22—2842no—
239C22—1424no—
240C22—1424no—
241C22—1424no—
242C22#41×D7112yes—
243C22C2×#1176586411yescharacteristic
244C22#1455617311yescharacteristic
245C22#1455617411yescharacteristic
246C22#1455617412yes—
247C4—42no—
248C4—42no—
249C4—41no—
250C4—41no—
251C4—2842no—
252C4—2842no—
253C4—2842no—
254C2—42no—
255C2—2844no—
256C2#1798658611yescharacteristic
257C2#1798658812yes—
258C2#1798667212yes—
259C2#1798667711yescharacteristic
260C2#1798674111yescharacteristic
subgroupG/H
center, FrattiniC23C23×D71
commutator, socleC23×C71C24
Fitting#397×C71C2
hypercenter#397D71
hypocenterC71#2736
LCSG ▷ C23×C71 ▷ C71
UCSC1 ◁ C23 ◁ #397

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
1. #17985926C2semidirect
2. #17985912C2semidirect
3. #17985911C2semidirect
4. #17985892C2semidirect
5. #17985891C2semidirect
6. #17985889C2semidirect
9. #397×C71C2semidirect
11. C22×Dic142C22semidirect
12. C2×C4⋊Dic71C22semidirect
21. C2×C4×Dic71C22semidirect
22. C71⋊#95C22semidirect
23. C71⋊#95C22semidirect
24. C71⋊#95C22semidirect
25. C71⋊#95C22semidirect
26. C2×C4⋊C4×C71C22semidirect
29. #95×C71C22semidirect
30. #95×C71C22semidirect
131. C71#2736semidirect
133. #397D71semidirect
144. C2×C4⋊C4C2×D71semidirect
160. #95C2×D71semidirect
161. #95C2×D71semidirect